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Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n (x-c)^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex.

A series of the form $$\sum_{n=0}^{\infty} a_n (x-c)^n$$ is called a power series, and can be used to expand functions. The centre $c$ is often $0$.

Power series for some common functions are: \begin{align} \frac{1}{1-x}&=\sum_{n=0}^\infty x^n\quad(|x|\lt1)\\\ e^x &= \sum_{n=0}^\infty\frac{x^n}{n!}\\\ \ln(1+x)&=\sum_{n=0}^\infty\frac{(-1)^nx^{n+1}}{n+1}\quad(|x|\lt1)\\\ \sin(x)&=\sum_{n=0}^\infty\frac{(-1)^n x^{2n+1}}{(2n+1)!}\\\ \cos(x)&=\sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} \end{align}

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